On Conway's Numbers and Games, the Von Neumann Universe, and Pure Set Theory

Abstract: We take up Dedekind's question ''Was sind und was sollen die Zahlen?'' (''What are numbers, and would should they be?''), with the aim to describe the place that Conway's (Surreal) Numbers and Games take, or deserve to take, in the whole of mathematics. Rather than just reviewing the work of Conway, and subsequent one by Gonshor, Alling, Ehrlich, and others, we propose a new setting which puts the theory of surreal numbers onto the firm ground of ''pure'' set theory. This approach is closely related to Gonshor's one by ''sign expansions'', but appears to be significantly simpler and clearer, and hopefully may contribute to realizing that ''surreal'' numbers are by no means surrealistic, goofy or wacky. They could, and probably should, play a central role in mathematics. We discuss the interplay between the various approaches to surreal numbers, and analyze the link with Conway's original approach via Combinatorial Game Theory (CGT). To clarify this, we propose to call pure set theory the algebraic theory of pure sets, or in other terms, of the algebraic structures of the von Neumann universe. This topic may be interesting in its own right: it puts CGT into a broad context which has a strong ''quantum flavor'', and where Conway's numbers (as well as their analogue, the nimbers) arise naturally.